Oblique Triangle Calculator
Solve non-right oblique triangles step-by-step using Law of Sines, Law of Cosines, and Heron's formula for all cases.
Non-Right Oblique Triangle Solver
Select input case and enter known sides/angles.
Oblique Triangle Solution Summary
Quick Summary
Our oblique triangle calculator law of sines cosines sss sas aas asa ssa steps tool solves non-right oblique triangles for all 5 input cases (SSS, SAS, ASA, AAS, SSA ambiguous case). It calculates missing side lengths $a, b, c$, interior angles $A, B, C$, surface area, perimeter, inradius $r$, and circumradius $R$.
How It Works: Law of Sines & Law of Cosines
An oblique triangle has no 90-degree right angle (can be acute or obtuse):
1. **Law of Sines:** $\a / \sin A = \b / \sin B = \c / \sin C$. Best for AAS, ASA, and SSA cases.
2. **Law of Cosines:** $c^2 = a^2 + b^2 - 2ab \cos C$. Best for SSS and SAS cases.
3. **Sum of Interior Angles:** $A + B + C = 180^\circ$.
4. **SSA Ambiguous Case:** Given 2 sides and a non-included angle $(a, b, A)$, height $h = b \sin A$ determines whether 0, 1, or 2 triangles exist.
5. **Heron's Area Formula:** $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$, where semiperimeter $s = \a+b+c / 2$.
Formula Explanation
Your non-right triangle calculations follow classical trigonometric laws:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for SSS with side lengths $a = 7, b = 8, c = 9$:
- Step 1 (Check Triangle Inequality): $7 + 8 = 15 > 9$, $7 + 9 = 16 > 8$, $8 + 9 = 17 > 7$. A valid non-degenerate triangle exists.
- Step 2 (Find Angle C via Law of Cosines): $\cos C = \7^2 + 8^2 - 9^2 / 2(7)(8) = \49 + 64 - 81 / 112 = \32 / 112 \approx 0.2857 \implies C = \arccos(0.2857) \approx \mathbf{73.40^\circ}$.
- Step 3 (Find Angle A via Law of Cosines): $\cos A = \8^2 + 9^2 - 7^2 / 2(8)(9) = \64 + 81 - 49 / 144 = \96 / 144 = \2 / 3 \approx 0.6667 \implies A = \arccos(0.6667) \approx \mathbf{48.19^\circ}$.
- Step 4 (Find Angle B): $B = 180^\circ - 48.19^\circ - 73.40^\circ = \mathbf{58.41^\circ}$.
- Step 5 (Compute Area via Heron's Formula): $s = \7+8+9 / 2 = 12$, $\text{Area} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx \mathbf{26.8328}$, delivering **A = 48.19°, B = 58.41°, C = 73.40°, Area = 26.83**!
Calculation Examples: Real-World Scenario Comparison
Compare input parameters across different geometric cases:
| Case / Inputs | Sides (a, b, c) | Angles (A, B, C) | Area | Triangles Found |
|---|---|---|---|---|
| SSS (7, 8, 9) | a=7, b=8, c=9 | A=48.19°, B=58.41°, C=73.40° | 26.8328 | 1 Unique |
| SAS (a=5, b=7, C=45°) | a=5, b=7, c=4.9701 | A=45.29°, B=89.71°, C=45.00° | 12.3744 | 1 Unique |
| ASA (A=40°, c=10, B=60°) | a=6.5270, b=8.7939, c=10 | A=40.00°, B=60.00°, C=80.00° | 28.2608 | 1 Unique |
| SSA Ambiguous (a=6, b=8, A=30°) | a=6, b=8, c=10.93 / 2.93 | B1=41.81°, B2=138.19° | 16.03 / 4.29 | 2 Ambiguous Solutions |
Benefits of Using the Oblique Triangle Calculator
Utilizing this calculator provides essential aviation flight vector drift correction, land boundary surveying, and civil engineering advantages:
- Aviation Flight Vector Navigation: Calculates ground speed and drift heading angles under crosswind conditions using SAS/AAS triangles.
- Land Surveying & Geodesy: Solves property boundary line lengths across rivers or obstacles where direct right-angle measurement is impossible.
- SSA Ambiguous Case Handling: Instantly detects whether 0, 1, or 2 valid geometric triangle solutions exist.
- Complete Radius Outputs: Calculates incircle and circumcircle radii essential for advanced geometry proofs.
Frequently Asked Questions (FAQ)
What is an oblique triangle?
An oblique triangle is any triangle that does NOT contain a 90-degree right angle. It can be acute (all angles < 90°) or obtuse (one angle > 90°).
What is the Law of Sines?
a / sin(A) = b / sin(B) = c / sin(C). It states that the ratio of a side to the sine of its opposite angle is constant for all 3 sides.
What is the Law of Cosines?
c^2 = a^2 + b^2 - 2ab * cos(C). It generalizes the Pythagorean theorem to any oblique triangle.
When should I use Law of Sines vs Law of Cosines?
Use Law of Sines for AAS, ASA, and SSA cases. Use Law of Cosines for SSS and SAS cases.
What is the SSA ambiguous case?
When given 2 sides and a non-included angle (SSA), there may be 0, 1, or 2 distinct valid triangles depending on whether side a is shorter than, equal to, or longer than altitude height h = b * sin(A).
What is Heron's formula for area?
Area = sqrt(s * (s - a) * (s - b) * (s - c)), where s = (a + b + c) / 2 is the semiperimeter.
What is the Triangle Inequality Theorem?
It states that for any valid triangle, the sum of lengths of any two sides must be strictly greater than the length of the remaining third side (a + b > c).
What is the inradius of an oblique triangle?
The inradius r is the radius of the circle inscribed inside the triangle, calculated as r = Area / s.
What is the circumradius of an oblique triangle?
The circumradius R is the radius of the circle circumscribed around all 3 vertices, calculated as R = (a * b * c) / (4 * Area).
What happens if the sum of two given angles is 180 degrees or more?
The calculator will display an error because the sum of all 3 interior angles must equal exactly 180 degrees.